Advances in Steiner Trees pp 117-135 | Cite as

# On Approximation of the Power-*p* and Bottleneck Steiner Trees

## Abstract

Many VLSI routing applications, as well as the facility location problem involve computation of Steiner trees with non-linear cost measures. We consider two most frequent versions of this problem. In the power-p Steiner problem the cost is defined as the sum of the edge lengths where each length is raised to the power *p >* 1. In the bottleneck Steiner problem the objective cost is the maximum of the edge lengths. We show that the power-p Steiner problem is MAX SNP-hard and that one cannot guarantee to find a bottleneck Steiner tree within a factor less than 2, unless P = NP. We prove that in any metric space the minimum spanning tree is at most a constant times worse than the optimal power-p Steiner tree. In particular, for *p =* 2, we show that the minimum spanning tree is at most 23.3 times worse than the optimum and we construct an instance for which it is 17.2 times worse. We also present a better approximation algorithm for the bottleneck Steiner problem with performance guarantee log_{2} n, where n is the number of terminals (the minimum spanning tree can be 2 log_{2} n times worse than the optimum).

## Keywords

Minimum Span Tree Steiner Tree Performance Ratio Steiner Point Steiner Tree Problem## Preview

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